Jump-induced mixed-mode oscillations through piecewise-affine maps

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:505

Article

Patsios, Yiorgos¹  Huzak, Renato¹  De Maesschalck, Peter¹  Popovic, Nikola^2,3 

[1] Hasselt Univ, Campus Diepenbeek,Agoralaan Gebouw D, B-3590 Diepenbeek, Belgium

[2] Univ Edinburgh, James Clerk Maxwell Bldg,Peter Guthrie Tait Rd, Edinburgh EH9 3FD, Midlothian, Scotland

[3] Maxwell Inst Math Sci, James Clerk Maxwell Bldg,Peter Guthrie Tait Rd, Edinburgh EH9 3FD, Midlothian, Scotland

关键词: Mixed-mode oscillation;    Relaxation oscillation;    Piecewise affine map;    Slow-fast system;    Geometric singular perturbation theory;   

DOI  :  10.1016/j.jmaa.2021.125641

来源: Elsevier

【 摘 要 】

Mixed-mode oscillations (MMOs) are complex oscillatory patterns in which large-amplitude oscillations (LAOs) of relaxation type alternate with small-amplitude oscillations (SAOs). MMOs are found in singularly perturbed systems of ordinary differential equations of slow-fast type, and are typically related to the presence of so-called folded singularities and the corresponding canard trajectories in such systems. Here, we introduce a canonical family of three-dimensional slow-fast systems that exhibit MMOs which are induced by relaxation-typ e dynamics, and which are hence based on a jump mechanism, rather than on a more standard canard mechanism. In particular, we establish a correspondence between that family and a class of associated one-dimensional piecewise affine maps (PAMs) which exhibit MMOs with the same signature. Finally, we give a preliminary classification of admissible mixed -mode signatures, verifying results of (Rajpathak, Pillai, and Bandyopahdyay (2012) [29]) in the process, and we illustrate our findings with numerical examples. (c) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

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